Haversine or Spherical Law of Cosines: Which Formula Is Best for Geo Proximity Calculations?

Deciphering Formulas for Geo Proximity Calculations

In the pursuit of implementing Geo proximity search functionality, selecting the appropriate formula can be daunting. Two primary options emerge: the Haversine Formula and the Spherical Law of Cosines. However, it's important to clarify that they are not completely interchangeable.

Haversine vs. Spherical Law of Cosines

The Haversine Formula uses a more robust approach, which makes it less susceptible to floating-point errors. However, for most practical applications, the Spherical Law of Cosines provides sufficient accuracy with approximately 15 significant figures.

Earth Shape Considerations

Both the Haversine Formula and the Spherical Law of Cosines assume a spherical Earth. For higher precision, Vicenty's formula is recommended, as it accounts for the Earth's ellipsoidal shape. This formula, however, is more computationally complex.

Geocentric vs. Geodetic Latitude

It's essential to note that the latitude used in both the Law of Cosines and the Haversine Formula is geocentric, which differs slightly from geodetic latitude. For a spherical Earth, these latitudes are equivalent.

Performance Considerations

In terms of computational efficiency, the Law of Cosines is the fastest, followed by the Haversine Formula, and finally Vicenty's formula. The latter, while most accurate, requires an iterative solution and is therefore slower.

Best Fit for Your Needs

The optimal choice of formula depends on the specific requirements of your application. If speed is paramount and Earth flatness can be assumed, a simplified formula (not described in the original text) can be utilized. If higher accuracy is desired, the Haversine Formula or Spherical Law of Cosines may suffice. For the most accurate results, especially over long distances, Vicenty's formula is recommended.

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